Hey, guys. So in this video, I want to start talking about springs and spring forces. Let's check it out. So when you push or pull against the spring with a force, let's call this force F_{a} an applied force, the spring pushes back. This is just Newton's 3rd law, action reaction. You push against something, and it pushes back against you. So the force of the spring will be the negative of the force that was applied on it, again, showcasing Newton's action reaction. That's why it's the same magnitude but in opposite direction.

This negative here simply signifies the opposite direction. What is new is that this also equals to −kx and I'll explain what k and x are in just a moment. Notice that there's another negative here, and that negative just means opposite direction. That negative, as you'll learn more about later, is almost a conceptual reminder that the force is in the opposite direction. In many calculations, we might disregard the negatives to simplify matters.

So, what are k and x? I'll start with x. X represents the deformation that the spring will experience, meaning if you push or pull against the spring, it will either be compressed (becoming shorter) or it will be stretched (becoming longer). However, x is not the actual length but the change in length. In terms of springs, it's not about how long it is, but how much longer or shorter it gets. X can be thought of as the absolute value of the final length minus the initial length. The reason it's taken as an absolute value is that the sign of x (negative or positive) doesn't impact its interpretation since it only shows the magnitude of change.

For example, if we have a spring attached to a wall, and the spring is uncompressed, its original length, with x equals 0, represents the relaxed position of the spring. In another scenario, portrayed here with the first drawing, I apply a force F_{a} to the right, resulting in the spring pushing back to the left with a force F_{s}. The difference between its original length and its new length is called its deformation, labeled as x or x_{1} for distinct situations.

K represents the spring's force constant, which essentially quantifies how stiff the spring is or how challenging it is to compress or stretch. The higher the k, the harder it is to deform the spring. K is typically measured in newtons per meter, as evident from the equation F = kx, where F is in newtons and x is in meters, rendering k as newtons per meter for dimensional consistency.

The force of a spring is fundamentally a restoring force, always attempting to bring the spring back to its original, unaltered length. If stretched or compressed, the spring exerts a force in the direction of returning to this equilibrium state.

Let's consider a practical example to clarify these concepts. Imagine a spring 1 meter long laid horizontally with one end fixed. If you exert a force of 50 newtons resulting in the spring stretching to a length of 1.2 meters, the change in spring length, x, is 0.2 meters. From this, if you wanted to determine the spring's k, knowing that the force and deformation are as stated, you could rearrange the formula to find k = 250 newtons per meter.

Now, addressing vertical springs, when a mass is attached to a spring and allowed to stretch by its weight, the spring will elongate until reaching a point of equilibrium where the upward spring force equals the downward gravitational force of the mass, described by the equation kx = mg. This balance signifies that the spring has reached a state where its force exactly counters the gravitational pull, maintaining the mass in a steady position.

In summary, understanding the mechanics of springs through the spring constant k and the deformation x, alongside recognizing the relationship between forces at play, forms the cornerstone of analyzing spring dynamics accurately. Let's remember this as we proceed with solving related problems in physics.